We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ℓψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type,...

The class of J-lattices was defined in the second author’s thesis. A subspace lattice on a Banach space X which is also a J-lattice is called a J- subspace lattice, abbreviated JSL. Every atomic Boolean subspace lattice, abbreviated ABSL, is a JSL. Any commutative JSL on Hilbert space, as well as any JSL on finite-dimensional space, is an ABSL. For any JSL ℒ both LatAlg ℒ and ${ℒ}^{\perp }$ (on reflexive space) are JSL’s. Those families of subspaces which arise as the set of atoms of some JSL on X are characterised...

We investigate the existence of higher order ℓ¹-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space $X=T\left[{(\theta ₙ,ₙ)}_{n=1}^{\infty }\right]$: (1) Every block subspace of X contains an $\ell ¹{-}_{\omega }$-spreading model, (2) The Bourgain ℓ¹-index $I_b\left(Y\right)=I\left(Y\right)>{\omega }^{\omega }$ for any block subspace Y of X, (3) $limₘlimsupₙ{\theta }_{m+n}/\theta ₙ>0$ and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions...

The aim of this work is to generalize lacunary statistical convergence to weak lacunary statistical convergence and $ℐ$-convergence to weak $ℐ$-convergence. We start by defining weak lacunary statistically convergent and weak lacunary Cauchy sequence. We find a connection between weak lacunary statistical convergence and weak statistical convergence.

On définit et étudie les espaces de Banach de type et de cotype $p$, $0\<p\le 2$. Cela permet de dire que, dès que certaines applications sont $q$-sommantes, elles sont aussi $r$-sommantes pour $r\le q$.

If $E=e_i$ and $F=f_i$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i,j}$ with norm $\left|\right|a_{i,j}{\left|\right|}_{E\left(F\right)}=\left|\right|{\sum }_k\left|\right|{\sum }_l{a}_{k,l}{f}_l\left|\right|e_k\left|\right|$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

We prove, among other things, that the space C[0,ω₂] has no countably norming Markushevich basis. This answers a question asked by G. Alexandrov and A. Plichko.

We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $\left(e_k\right)$ is said to be subsequentially minimal if for every normalized block basis $\left(x_k\right)$ of $\left(e_k\right)$, there is a further block basis $\left(y_k\right)$ of $\left(x_k\right)$ such that $\left(y_k\right)$ is equivalent to a subsequence of $\left(e_k\right)$. Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of mixed Tsirelson...